A devil's staircase from rotations and irrationality measures for Liouville numbers

AbstractFrom Sturmian and Christoffel words we derive a strictly increasing function Δ:[0,∞) → . This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of Δ distinguishes some irrationality measures of real numbers.

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Univoque bases of real numbers: Local dimension, Devil's staircase and isolated points

Advances in Applied Mathematics ◽

10.1016/j.aam.2020.102103 ◽

2020 ◽

Vol 121 ◽

pp. 102103

Author(s):

Derong Kong ◽

Wenxia Li ◽

Fan Lü ◽

Zhiqiang Wang ◽

Jiayi Xu

Keyword(s):

Local Dimension ◽

Real Numbers ◽

Devil's Staircase ◽

Devil’S Staircase ◽

Isolated Points

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NMR spectra and relaxation in incommensurate systems in the presence of a devil's staircase

Journal de Physique ◽

10.1051/jphys:019850046070120500 ◽

1985 ◽

Vol 46 (7) ◽

pp. 1205-1209 ◽

Cited By ~ 5

Author(s):

R. Blinc ◽

S. Žumer ◽

D.C. Ailion ◽

J. Nicponski

Keyword(s):

Nmr Spectra ◽

Devil's Staircase ◽

Devil’S Staircase

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Fractal Dimension and Self-Similarity of the Devil's Staircase in a Josephson-Junction Simulator

Physical Review Letters ◽

10.1103/physrevlett.52.480 ◽

1984 ◽

Vol 52 (6) ◽

pp. 480-480 ◽

Cited By ~ 53

Author(s):

W. J. Yeh ◽

Da-Ren He ◽

Y. H. Kao

Keyword(s):

Fractal Dimension ◽

Josephson Junction ◽

Self Similarity ◽

Devil's Staircase ◽

Devil’S Staircase

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Phenomenology of a Devil's Staircase

Crystallography Reviews ◽

10.1080/08893119208032971 ◽

1992 ◽

Vol 3 (2) ◽

pp. 231-250

Author(s):

D. G. Sannikov

Keyword(s):

Devil's Staircase ◽

Devil’S Staircase

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Devil's-staircase behavior of a simple spin model

Physical Review B ◽

10.1103/physrevb.24.2744 ◽

1981 ◽

Vol 24 (5) ◽

pp. 2744-2750 ◽

Cited By ~ 31

Author(s):

E. B. Rasmussen ◽

S. J. Knak Jensen

Keyword(s):

Spin Model ◽

Devil's Staircase ◽

Devil’S Staircase

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Distribution of rational points on the real line

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700013008 ◽

1973 ◽

Vol 15 (2) ◽

pp. 243-256 ◽

Cited By ~ 1

Author(s):

T. K. Sheng

Keyword(s):

Algebraic Number ◽

Rational Number ◽

Real Line ◽

Rational Points ◽

The Other ◽

Liouville Numbers ◽

The Real ◽

Other Hand ◽

Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

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